121 T/O Mins: These are specified in your OPSPECS and can vary between airlines. Typically it goes like this:
1. If the 10-9 does NOT publish higher-than-standard, you can always use standard.
2. If you can use standard, then your OPSPEC will typically allow you to use your lower-than-standard (5-5-5, 6-6-6, or whatever your airline has).
3. If the 10-9 publishes HIGHER than standard, then you are limited to the 10-9, you cannot use OPSPEC lower-than-standard.
Basically, if standard is published or implied, you can use OPSPEC mins. If higher-than-standard is published, you must use the published mins.
Note: If the 10-9 publishes different values for different climb gradients, you will base your determination on the mins which apply to your climb gradient.
ALT Mins: These are normally calculated based on an OPSPEC formula using the CIG and VIS mins for the approaches to the ALT airport. Too complicated to discuss here, but OPSPEC will overide the standard 600-2/800-2. The formula will produce mins higher than basic approach mins.
The 10-9 ALT mins don't count for 121, except if the IAP says "N/A" under alternate mins for your category, you cannot use that airport as an ALT.
At the airlines I have been at, ALT min calculations worked the same for both T/O and DEST alternates.